Isolated singularities of mappings with the inverse Poletsky inequality

E.A. Sevost'yanov

doi:10.30970/ms.55.2.132-136

E.A. Sevost'yanov Zhytomyr Ivan Franko State University, Zhytomyr, Ukraine; Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Slavyansk, Ukraine

DOI: https://doi.org/10.30970/ms.55.2.132-136

Keywords: quasiconformal mappings;, mappings with bounded and finite distortion;, equicontinuity;, moduli of families of paths

Abstract

The manuscript is devoted to the study of mappings
with finite distortion, which have been actively studied recently.
We consider mappings satisfying the inverse Poletsky inequality,
which can have branch points. Note that mappings with the reverse
Poletsky inequality include the classes of con\-for\-mal,
quasiconformal, and quasiregular mappings. The subject of this
article is the question of removability an isolated singularity of a
mapping. The main result is as follows. Suppose that $f$ is an open
discrete mapping between domains of a Euclidean $n$-dimensional
space satisfying the inverse Poletsky inequality with some
integrable majorant $Q.$ If the cluster set of $f$ at some isolated
boundary point $x_0$ is a subset of the boundary of the image of the
domain, and, in addition, the function $Q$ is integrable, then $f$
has a continuous extension to $x_0.$ Moreover, if $f$ is finite at
$x_0,$ then $f$ is logarithmic H\"{o}lder continuous at $x_0$ with
the exponent $1/n.$

References

E.A. Sevost’yanov, S.A. Skvortsov, On mappings whose inverse satisfy the Poletsky inequality, Ann. Acad. Scie. Fenn. Math., 45 (2020), 259–277.

V. Ya. Gutlyanskii, V. I. Ryazanov, E. Yakubov, The Beltrami equations and prime ends, Ukr. Mat. Visn., 12 (2015), No1, 27–66; transl. J. Math. Sci. (N.Y.), 210 (2015), No1, 22–51.

O. Martio, S. Rickman, J. V¨ais¨al¨a, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A1, 448 (1969), 1–40.

O. Martio, S. Rickman, J. V¨ais¨al¨a, Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A1, 488 (1971), 1–31.

O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in modern mapping theory, Springer Monographs in Mathematics, Springer, New York etc., 2009.

O. Martio, U. Srebro, Periodic quasimeromorphic mappings in Rn, J. d’Anal. Math., 28 (1975), No1,20–40.

S. Rickman, Quasiregular mappings, Springer-Verlag, Berlin, 1993.

J. V¨ais¨al¨a, Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Math., 229, Springer–Verlag, Berlin etc., 1971.

K. Kuratowski, Topology, V.2, Academic Press, New York–London, 1968.

E. Sevost’yanov, Towards a theory of removable singularities for maps with unbounded characteristic of quasi-conformity, Izv. Math., 74 (2010), No1, 151–165.

E. Sevost’yanov, S. Skvortsov, O. Dovhopiatyi, On nonhomeomorphic mappings with the inverse Poletsky inequality, Ukr. Mat. Visnyk, 17 (2020), No3, 414–436. (in Ukrainian)